Elsevier

Composite Structures

Volume 72, Issue 1, January 2006, Pages 10-18
Composite Structures

Supersonic flutter characteristics of functionally graded flat panels including thermal effects

https://doi.org/10.1016/j.compstruct.2004.10.007Get rights and content

Abstract

In the present work, the influence of thermal environment on the supersonic flutter behavior of flat panels made of functionally graded materials is investigated using the finite element procedure. The structural formulation is based on first-order shear theory and material properties are assumed to be temperature dependent and graded in the thickness direction according to power law distribution in terms of the volume fractions of the constituents. The aerodynamic force is evaluated by considering the first-order high Mach number approximation to linear potential flow theory. The solutions for the complex eigenvalue problem, formulated based on Lagrange’s equation of motion, are obtained using the standard eigenvalue algorithm. The variation of critical aerodynamic pressure is brought out considering different parameters such as plate thickness, aspect ratio, power law index of functionally graded materials, temperature-dependent material properties, damping and boundary conditions. The influence of aerodynamic damping and thermal gradient on the flutter behavior of functionally graded plates is also highlighted.

Introduction

During the last few decades, an increased effort towards integrating advanced composite materials in the construction of aerospace structures was manifested. Due to their well-known advantages related to strength-to- and stiffness-to-weight ratios, and the possibility of tailoring their anisotropic properties in optimizing their structural responses, their engineering applications will even increase in the years ahead. However, as these structures may yield the premature failure by fatigue or by the decay of stiffness characteristics of laminated structures because of delaminations and chemically unstable matrix and lamina adhesives, it is necessary to explore new structural solutions to overcome the occurrence of such catastrophic events during their service life. The recent emergence of functionally graded materials [1], [2] that combine the best properties of metals and ceramics, can be thought of an alternative solution for certain class of aerospace structures exposed to a high temperature environment. Furthermore, in practice, the use of these materials in aerospace industries has necessitated to understand the dynamic characteristics of functionally graded structures.

It is observed from the existing literature that most of the research work on functionally graded plates have been restricted to thermal stress and buckling analyses. Limited work has been reported on dynamic analysis of functionally graded plates [3], [4], [5], [6], [7]. Cheng and Batra [3] have studied the steady state vibration of a simply supported functionally graded polygonal plate with temperature independent material properties. Recently, Yang and Shen [4] have examined the vibration characteristic and transient response of functionally graded plates in thermal environments. He et al. [5] have presented the vibration control of FGM thin plate with integrated piezoelectric sensors and actuators. An attempt is also made in the literature to bring out the nonlinear dynamic behavior of plates made of functionally graded materials [6], [7]. Praveen and Reddy [6] have analyzed the dynamic response of functionally graded ceramic metal plates through the finite element method whereas Yang, et al. [7] investigated the large amplitude vibration of thermo-electro-mechanically stressed FGM laminated plates. However, the studies pertaining to dynamic instability of functionally graded structures is rather sparsely treated in the literature compared to vibration analysis. Ref. [8] deals with the parametric resonance of functionally graded plates subjected to periodic axial load.

The flutter phenomenon is one such dynamic instability problem encountered in the flight of aerospace vehicles. It is the self-excited oscillation of the external skin of a flight vehicle when exposed to airflow along its surface and there is a critical pressure above which the panel motion becomes unstable and grows exponentially with time. This study pertaining to isotropic and laminated composite cases have received considerable attention in the literature and are reviewed by Dowell [9] and recently by Bismark-Nasr [10], [11]. However, to the best of authors’ knowledge, the investigation of the flutter behavior of functionally graded flat panels has not been yet accomplished in the literature. As the use of panels made of functionally graded materials is expected to increase in the coming years in the aerospace industry, it is worth investigating the dynamic stability characteristics of such structures exposed to aerodynamic flow.

In this paper, an eight-noded C0 shear flexible quadrilateral plate-bending element developed recently [12], [13] is extended to analyze the supersonic panel flutter behavior of functionally graded plates. Since the element is based on field consistency approach, exact integration is applied for calculating various energy terms. The aerodynamic force is evaluated considering the first-order high Mach number approximation to linear potential flow theory. The QR algorithm is employed for the solution of complex eigenvalue problem. The formulation developed herein is validated with the available solutions. A detailed parametric study has been carried out to bring out the influences of the plate thickness, aspect ratio, power law index, damping and boundary conditions on the flutter behavior of FGM plates. The influence of thermal environment on the flutter behavior is also evaluated.

Section snippets

Theoretical development and formulation

A functionally graded rectangular plate (length a, width b, and thickness h) made of a mixture of ceramics and metals is considered with the coordinates x, y along the in-plane directions and z along the thickness direction. The material in top surface (z = h/2) of the plate and in bottom surface (z = h/2) of the plate is ceramic and metal, respectively. The effective material properties P, such as Young’s modulus E, and thermal expansion coefficient α, can be written as [6]P=PcVc+PmVmwhere Pc and

Results and discussion

Here, an eight-noded shear flexible quadrilateral plate element, based on field-consistency approach [12], [13] is extended to analyse the flutter characteristics pertaining to functionally graded plates. It has five nodal degrees of freedom associated to u0, v0, w, θx and θy at eight nodes in the element. The formulation includes transverse shear deformation, and in-plane and rotary inertia effects. Since the element is derived using field-consistency approach, full integration scheme is

Conclusions

The supersonic flutter of functionally graded flat panels has been analyzed based on first-order shear deformation theory through finite element approach. The material properties are assumed to vary through the thickness direction based on power law distribution. The aerodynamic force is accounted for assuming the first-order high Mach number approximation to linear potential flow theory. The effectiveness of volume fraction index, aspect and thickness ratios, boundary conditions and thermal

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